That may not be easy. In the House, Ohio's John Boehner will continue as GOP leader, and in the Senate, Kentucky's Mitch McConnell will serve another stint as Republican leader. At the Republican National Committee, six people are vying for the chairmanship, including incumbent Mike Duncan.

Bush said Republicans should not despair, even though, he said, "We got whipped in 2008."

"Everybody said the party was wiped out," Bush recalled. "And then a whole new wave of Republicans ran," including his father, who won a Houston-based congressional seat. "Same thing will happen," the president predicted.

"But," he added, "it's very important for our party not to narrow its focus, not to become so inward-looking that we drive people away from a philosophy that is compassionate and decent."

Just be sure, he advised, to be open-minded. "We shouldn't have litmus tests as to whether or not you can be a Republican," he said.

And, he added, "We should be open-minded about big issues like immigration reform, because if we're viewed as anti-somebody -- in other words, if the Party is viewed as anti-immigrant, then another fellow may say, well, if they're against the immigrant, they may be against me."

Republicans are divided on immigration; Bush pushed for broad reforms that would help create a path to citizenship for many immigrants, while others in his party wanted to stress enforcement and little else.